In Fig. REF , we present the quasienergy (Floquet) spectrum and gap functions of the first model in Sec.  and its square-root descendant
under both the PBC (gray dots in the background) and OBC. The quasienergies and gap functions of the parent model
in Figs. REF (a) and REF (c) are reproduced
from Ref. [1]}. A clear distinction between the spectrum under PBC (in gray dots) and OBC (in blue dots) can be observed especially around the phase transition points,
implying the presence of NHSE in the system. To retrieve the bulk-edge
correspondence, a pair of open-boundary winding numbers \((\nu _{0},\nu _{\pi })\) 
is introduced in Ref. [1]}, which correctly counts the number
of zero- and \(\pi \) -quasienergy edge modes \(n_{0}\)  and \(n_{\pi }\) 
in the parent model through the relation \((n_{0},n_{\pi })=2(|\nu _{0}|,|\nu _{\pi }|)\) .
Here we use \(n_{E}\)  to denote the number of edge states at the quasienergy
\(E\) . According to our square-root procedure, the edge modes at the quasienergies
\(E=0,\pm \pi \)  (\(E=\pm \pi /2\) ) are generated by taking the square-root
of the \(\pi \)  (zero) Floquet edge modes. Therefore, we arrive at the
following bulk-edge correspondence for the square-root non-Hermitian
FTIs described by \(U_{1/2}\) , i.e.,
\(n_{\pi /2}=2|\nu _{0}|,\qquad n_{0}=n_{\pi }=2|\nu _{\pi }|,\) 
